That depends on what weight you put on “quality of students” and “quality of faculty/teaching”.
Probably everyone thinks that the quality of faculty is important; they’re teaching the students.
The weight we put on the importance of student quality is likely more variable. The more you think student quality is important, the more likely you are to believe that selectivity is important.
I started to write a response, but now my mod self is taking over. Let’s drop the discussion of rankings and what makes up the quality of a school and get back to the OP’s situation. I know it is hard not to respond (grrr…) but da rules are da rules.
Case Western maybe?
When they say UChicago is unique, it’s for a very good reason. In many ways, it is singular–academics, the setting and history, the nature and type of student body, curriculum, etc.
I think Johns Hopkins can be similar. (Dominant undergrad vibe is more pre-professional, but there’s a pretty robust intellectual culture and faculty/grad students are welcoming to undergrads who share those values.) No core, but great humanities departments. And an engineering school. Slightly higher acceptance rate and lower median scores than U of C.
Not including similarly competitive schools, below are some similar schools that are a little easier to get into and placed in major cities:
NYU
Boston College
Georgetown
Northeastern University
John Hopkins
Loyola Chicago
UIChicago
Fordham
I have to disagree with @DetroitLeper. While they may have similar “geek chic” social cultures, I would say that Rochester and Chicago are at opposite ends of the continuum when it comes to curriculum. Rochester is wonderfully “fluid” in terms of letting students create their own curricula through the use of the “cluster” approach. Chicago, on the other hand, is pretty rigid re. core requirements (which, if I recall correctly, can account for about 1/3 of the hours needed for graduation). I see merits to both approaches. On one hand, Rochester lets students really immerse themselves in what interests them. But as someone who works with a lot of Chicago alums, I’ve come to recognize how a strong core curriculum leads to grads with a recognizable skill set that I admire.
@EllieMom I admit being unaware of how curriculum works with the other UAA schools. I was merely suggesting somewhat similar campus cultures and pointing out a few schools that I had liked for myself. I did not attempt to tackle the other considerations that OP asked for.
If looking for an intellectual student focus, U Chicago is considered most similar to Swarthmore which many consider to be similar to Carleton. Carleton has excellent sciences, but no engineering.